# Parallelogram and trapezoid

*Parallelogram. Properties of a parallelogram.*

Signs of a parallelogram. Rectangle. Rhombus.

Square. Trapezoid. Isosceles trapezoid.

Midline of a trapezoid and a triangle.

Signs of a parallelogram. Rectangle. Rhombus.

Square. Trapezoid. Isosceles trapezoid.

Midline of a trapezoid and a triangle.

** Parallelogram **( ABCD, Fig.32 ) is a quadrangle, opposite sides of which are two-by-two parallel.

Any two opposite sides of a parallelogram are called *bases*, a distance between them is called a *height* ( BE, Fig.32 ).

*Properties of a parallelogram.*

1. *Opposite sides of a parallelogram are equal *( AB = CD, AD = BC ).

2. *Opposite angles of a parallelogram are equal* ( A =

C, B =

D ).

3. *Diagonals of a parallelogram are divided in their intersection point into two *

( AO = OC, BO = OD ).

4. *A sum of squares of diagonals is equal to a sum of squares of four sides*:

AC² + BD²

= AB² + BC²

+ CD² + AD² .

*Signs of a parallelogram.*

A quadrangle is a parallelogram, if one of the following conditions takes place:

1. *Opposite sides are equal two-by-two *( AB = CD, AD = BC ).

2. *Opposite angles are equal two-by-two* (

A =

C,

B =

D ).

3. *Two opposite sides are equal and parallel *( AB = CD, AB || CD ).

4. *Diagonals are divided in their intersection point into two* ( AO = OC, BO = OD ).

*Rectangle.*

If one of angles of parallelogram is right, then all angles are right (why ?). This parallelogram is called a *rectangle *( Fig.33 ).

__Main properties of a rectangle.__

Sides of rectangle are its heights simultaneously.

*Diagonals of a rectangle are equal:* AC = BD.

*A square of a diagonal length is equal to a sum of squares of its sides’ lengths* ( see above Pythagorean theorem ):

= AD² + DC².

** Rhombus. **If all sides of parallelogram are equal, then this parallelogram is called a

*rhombus*( Fig.34 ) .

*Diagonals of a rhombus are mutually perpendicular* ( AC BD ) *and divide
its angles into two* ( DCA = BCA,

ABD = CBD etc. ).

** Square **is a parallelogram with right angles and equal sides ( Fig.35 ).

*A square is a particular case of a rectangle and a rhombus*

simultaneously; so, it has all their above mentioned properties.

simultaneously; so, it has all their above mentioned properties.

** Trapezoid **is a quadrangle, two opposite sides of which are parallel (Fig.36).

Here AD **||** BC. Parallel sides are called *bases* of a trapezoid, the two others ( AB and CD ) – *lateral sides.* A distance between bases

(BM) is a * height. * The segment EF, joining midpoints E and F of the lateral sides, is called a *midline* of a trapezoid.

*A midline of a trapezoid is equal to a half-sum of bases:*

*and parallel to them: *EF || AD and EF || BC.

A trapezoid with equal lateral sides ( AB = CD ) is called an *isosceles* trapezoid. *In an isosceles trapezoid angles by each base, are equal *

( A = D,

B = C ).

A parallelogram can be considered as a particular case of trapezoid.

** Midline of a triangle** is a segment,

*joining midpoints of lateral sides*of a triangle.

*A midline of a triangle is equal to half*

of its base and parallel to it.This property follows from the previous part, as triangle can be considered as a limit case (“degeneration”) of a

of its base and parallel to it.

trapezoid, when one of its bases transforms to a point.